## Optical parametric oscillations in isotropic photonic crystals

Optics Express, Vol. 12, Issue 5, pp. 823-828 (2004)

http://dx.doi.org/10.1364/OPEX.12.000823

Acrobat PDF (195 KB)

### Abstract

We investigate optical parametric oscillations via four-wave mixing in a dielectric photonic crystal. Using a fully vectorial 3D time-domain approach, including both dispersion and Kerr nonlinear polarization, we analyze the response of an inverted opal. The results demonstrate the feasibility of parametric sources in isotropic media arranged in photonic band-gap geometries.

© 2004 Optical Society of America

1. S. Barland*et al*., “Cavity solitons as pixels in semiconductor microcavities,” Nature **419**, 699–702 (2002). [CrossRef] [PubMed]

2. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature **415**, 621–623 (2002). [CrossRef] [PubMed]

3. A. Yariv and W. H. Louisell, “Theory of the Optical Parametric Oscillator,” IEEE J. Quantum Electron. **QE-2**, 418–424 (1966). [CrossRef]

4. B. Crosignani, P. D. Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. **78**, 237–239 (1990). [CrossRef]

*ω*can generate two frequencies

*ω*

_{±}, with

*ω*

_{+}+

*ω*

_{-}=2

*ω*, via parametric oscillation at powers above the threshold:

*Q,Q*

_{+},

*Q*

_{-}, the cavity Q-factors at

*ω,ω*

_{+}and

*ω*

_{-}respectively, and g the pertinent three-dimensional overlap integral between the involved mode profiles and the spatial distribution of the nonlinearity. High Qs obviously favor parametric effects.

2. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature **415**, 621–623 (2002). [CrossRef] [PubMed]

6. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

7. S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

10. P. Tran, “Photonic-band-structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B **52**, 10673–10676 (1995). [CrossRef]

11. V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E **63**, 027602 (2001). [CrossRef]

12. S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B **19**, 2241–2249 (2002). [CrossRef]

13. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E **66**, 55601 (2002). [CrossRef]

14. J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, “Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,” IEEE J. Quantum Electron. **35**, 1168–1175 (1999). [CrossRef]

15. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Optics Express **10**, 670–684 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670 [CrossRef] [PubMed]

*a*

_{±}the amplitudes of the generated frequencies (normalized to unit energy), above threshold:

*p*measures the excess pumping, which is transferred to other generated frequencies, as in standard optical parametric oscillators (OPO) (see Refs. [3

3. A. Yariv and W. H. Louisell, “Theory of the Optical Parametric Oscillator,” IEEE J. Quantum Electron. **QE-2**, 418–424 (1966). [CrossRef]

*Pin*is the power excitation at the pump frequency

*ω*,

*ω*+

*ω*

_{+}-

*ω*

_{-}). With reference to the generation of a specific ω2, the energy stored in the cavity is:

*ρ*(

*ω*) of the cavity [9]. In deriving Eq. (3) we considered that, if

*Q*

_{2}≡

*Q*(

*ω*

_{2}) is sufficiently high, the relevant modes are those in proximity of

*ω*

_{2}.

19. Z. Y. Li and Y. Xia, “Full vectorial model for quantum optics in three-dimensional photonic crystals,” Phys. Rev. A **63**, 043817 (2001). [CrossRef]

*ρ*(

*ω*

_{2}). Hence, owing to FWM, we expect the output spectrum at high fluences to resemble the DOS, thus enabling the

*nonlinear spectroscopy*of PC-microresonators. This holds valid also for those modes excited in the non-stationary regime, i.e., not pumped by a specific frequency-mixing process and relying on transient parametric fluorescence. Since their dynamics is strongly linked to the time-scale of the nonlinearity, they need be modeled in conjunction with a non-instantaneous nonlinear response. This picture of parametric wave mixing in a PC microcavity gets even more involved when considering not only the degenerate interaction (2

*ω=ω*

_{+}+

*ω*

_{-}), but also processes of the type

*ω=ω*

_{a}+

*ω*

_{b}+

*ω*

_{c}. Hence a comprehensive numerical approach is required.

20. J. L. Young and R. O. Nelson, “A Summary and Sistematic Analysis of FDTD Algorithms for Linearly Dispersive Media,” IEEE Antennas Propagat. Mag. **43**, 61–77 (2001). [CrossRef]

**P**in the regions where material is present:

20. J. L. Young and R. O. Nelson, “A Summary and Sistematic Analysis of FDTD Algorithms for Linearly Dispersive Media,” IEEE Antennas Propagat. Mag. **43**, 61–77 (2001). [CrossRef]

*f*(

*P*), with

*f*(

*P*)=1 describing a linear single-pole dispersive medium (

*P*

^{2}=

**P**·

**P**). For an isotropic material, we used

*f*(

*P*)=[1+(

*P/P*

_{0})

^{2}]

^{-3/2}as in Ref. [23

23. J. Koga, “Simulation model for the effects of nonlinear polarization on the propagation of intense pulse laser,” Opt. Lett. **24**, 408–410 (1999). [CrossRef]

*P*

_{0}(

*ε*

_{s}-1)

^{3}

*ε*

_{s}

*n*

_{2}) a measure of the nonlinearity linked to the Kerr coefficient

*n*

_{2}. This provides a simple way to describe non-instantaneous FWM and higher order nonlinearities, [24] distinguishing our approach from previous ones (see e.g., Ref. [25

25. R. M. Joseph and A. Taflove, “FDTD Maxwell’s Equations Models for Nonlinear Electrodynamics and Optics,” IEEE Trans. Antennas Propagat. **45**, 364–374 (1997). [CrossRef]

*P/P*

_{0},

*f*produces a Kerr response, but compared to the standard Kerr

*f*(

*P*)=1+

*χ*

^{P2}, the resulting algorithm is stable even near the Courant limit [22]. This approach accounts for all the FWM interactions of the type

*ω=ω*

_{a}+

*ω*

_{b}+

*ω*

_{c}, but also higher order processes (e.g.,

*χ*

^{(5)}), in principle up to an infinite number of frequencies, the leading effect being cubic.

^{1}We computed the response of an inverted opal PC: a Face-Centered-Cubic (FCC) lattice (of period

*a*) of air-spheres (radius

*r*=0.3535

*a*) embedded in a dielectric (see inset of Fig. 1). The FCC is among the simplest structures admitting a complete photonic-band-gap (PBG) [26

26. S. John and K. Busch, “Photonic Bandgap Formation and Tunability in Certain Self-Organizing Systems,” J. Lightwave Technol. **17**, 1931–1943 (1999). [CrossRef]

27. A. Blanco*et al*., “Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,” Nature **405**, 437–440 (2000). [CrossRef] [PubMed]

*µm*

^{3}, was placed in air and excited by a 2

*µm*-waist linearly y-polarized Gaussian beam, numerically implemented through a total field/scattered field layer [22]. The FCC lattice, for an index 3.5 as in Si or GaAs, has a complete band-gap around the normalized frequency

*a/λ*=0.8, with λ the wavelength. To obtain a gap near λ=1500

*nm*we chose

*a*=1200

*nm*, and the parameters of the single pole dispersion were taken as

*ε*

_{s}≅11.971,

*ω*

_{0}=1.1×10

^{16}and

*γ*

_{0}=2×10

^{5}(MKS units), yielding an index≅3.5. An

*n*

_{2}=1.5×10

^{-17}

*m*

^{2}

*W*

^{-1}[28

28. J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear optical properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. **33**, 341–348 (1997). [CrossRef]

*P*

_{0}≅1

*Cm*

^{-2}. The integration domain was discretized with

*dx*≅

*dy*≅

*dz*≅30

*nm*and temporal steps

*dt*=0.02

*fs*, allowing more than 40 points at each wavelength and runs with 30000 steps in time (the spectral resolution is of the order of 10

*nm*at λ=1500

*nm*).

26. S. John and K. Busch, “Photonic Bandgap Formation and Tunability in Certain Self-Organizing Systems,” J. Lightwave Technol. **17**, 1931–1943 (1999). [CrossRef]

29. R. Wang, X. H. Wang, B. Y. Gu, and G. Z. Yang, “Local density of states in three-dimensional photonic crystals: Calculation and enhancement effects,” Phys. Rev. B **67**, 155114 (2003). [CrossRef]

^{2}A low-power (1

*n*

^{W}) single-cycle pulse [30

30. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E **64**, 056625 (2001). [CrossRef]

*X*direction and the

*E*

^{y}component of the transmitted signal was analyzed just after it. The resulting spectrum is shown in Fig. 1, where peaks correspond to concentrations of states (taken aside the low-frequency oscillations, due to the finiteness of the structure [9]) compatibly with symmetry constraints. The band structure of this medium encompasses a PBG around 1500

*nm*and a pseudo-gap around 2400

*nm*. Using these results, in order to cw pump the nonlinear parametric processes we picked λ≅1336

*nm*corresponding to

*a/λ*=0.898, i.e. close to a state by the upper edge of the PBG (in frequency), as marked by the star in Fig. 1. The 600

*f s*quasi-cw excitation was realized by an

*mnm pulse*with spectrum well peaked at the carrier frequency [30

30. R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E **64**, 056625 (2001). [CrossRef]

*mnm pulse*is a sinusoidal signal at the carrier (

*ω*), with a constant amplitude for n periods, and smooth trailing and tailing edges of

*m*periods. It allows the implementation of a quasi-CW (narrow-band) signal without a spurious frequency content. However, since simulations run for hundred fs, the results could be observed with pulses of similar duration.

*E*

^{y}spectral density obtained in a low-symmetry point at the center of the PC, for a

*y*-polarized pump propagating along the Γ

*X*direction. The insets show the generated frequencies as the input power is increased. Large output bandwidths are attained, with no oscillations at frequencies within the PBG, a smooth profile in the large wavelength region and several peaks above the PBG upper-edge. Each peak corresponds to a region with a large DOS [26

26. S. John and K. Busch, “Photonic Bandgap Formation and Tunability in Certain Self-Organizing Systems,” J. Lightwave Technol. **17**, 1931–1943 (1999). [CrossRef]

10. P. Tran, “Photonic-band-structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B **52**, 10673–10676 (1995). [CrossRef]

11. V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E **63**, 027602 (2001). [CrossRef]

**17**, 1931–1943 (1999). [CrossRef]

*ω=ω*

_{+}+

*ω*

_{-},

*ω*

_{+}and

*ω*

_{-}need not be symmetrically located with respect to the pump. Such asymmetries, as well as the spectrum dependence on power, were previously observed through cubic parametric amplification in fibers,[16] and are clearly accentuated in dispersive PC microcavities.

*P*=1.2

*MW*, as graphed in Fig. 3. Large wavelengths are substantially emitted due to lower Q-factors, while lower λ’s, although more efficiently generated (see Fig. 2), are not well out-coupled. Hence, the generation of a specific frequency would require an appropriate out-coupling.

31. M. J. A. d. Dood, A. Polmand, and J. G. Flerning, “Modified spontaneous emission from erbium-doped photonic layer-by-layer crystals,” Phys. Rev. B **67**, 115106 (2003). [CrossRef]

32. K. Banaszek and P. L. Knight, “Quantum interference in three-photon down-conversion,” Phys. Rev. A **55**, 2368 (1997) [CrossRef]

## Acknowledgements

## Footnotes

1 | The program runs on the IBM-SP4 system at the Italian Interuniversity Consortium for Advanced Calculus (CINECA), as well as on the NOMAD-BEOWULF cluster at NOOEL |

2 | It must also be noticed that -at best- the frequency domain analysis is of order N log(N), while the time-domain is of order N, with N the dimension of the problem. |

## References and links

1. | S. Barland |

2. | S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature |

3. | A. Yariv and W. H. Louisell, “Theory of the Optical Parametric Oscillator,” IEEE J. Quantum Electron. |

4. | B. Crosignani, P. D. Porto, and A. Yariv, “Coupled-mode theory and slowly-varying approximation in guided-wave optics,” Opt. Commun. |

5. | H. A. Haus, |

6. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

7. | S. John, “Strong Localization of Photons in Certain Disordered Dielectric Superlattices,” Phys. Rev. Lett. |

8. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, |

9. | K. Sakoda, |

10. | P. Tran, “Photonic-band-structure calculation of material possessing Kerr nonlinearity,” Phys. Rev. B |

11. | V. Lousse and J. P. Vigneron, “Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,” Phys. Rev. E |

12. | S. F. Mingaleev and Y. S. Kivshar, “Nonlinear transmission and light localization in photonic-crystal waveguides,” J. Opt. Soc. Am. B |

13. | M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E |

14. | J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, “Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,” IEEE J. Quantum Electron. |

15. | K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Optics Express |

16. | G. P. Agrawal, |

17. | J.-P. Fève, B. Boulanger, and J. Douady, “Specific properties of cubic optical parametric interactions compared to quadratic interactions,” Phys. Rev. A |

18. | M. Bahl, N.-C. Panoiu, and R. Osgood Jr.“Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,” Phys. Rev. E |

19. | Z. Y. Li and Y. Xia, “Full vectorial model for quantum optics in three-dimensional photonic crystals,” Phys. Rev. A |

20. | J. L. Young and R. O. Nelson, “A Summary and Sistematic Analysis of FDTD Algorithms for Linearly Dispersive Media,” IEEE Antennas Propagat. Mag. |

21. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. |

22. | A. Taflove and S. C. Hagness, |

23. | J. Koga, “Simulation model for the effects of nonlinear polarization on the propagation of intense pulse laser,” Opt. Lett. |

24. | R. W. Boyd, |

25. | R. M. Joseph and A. Taflove, “FDTD Maxwell’s Equations Models for Nonlinear Electrodynamics and Optics,” IEEE Trans. Antennas Propagat. |

26. | S. John and K. Busch, “Photonic Bandgap Formation and Tunability in Certain Self-Organizing Systems,” J. Lightwave Technol. |

27. | A. Blanco |

28. | J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, “The nonlinear optical properties of AlGaAs at the half band gap,” IEEE J. Quantum Electron. |

29. | R. Wang, X. H. Wang, B. Y. Gu, and G. Z. Yang, “Local density of states in three-dimensional photonic crystals: Calculation and enhancement effects,” Phys. Rev. B |

30. | R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E |

31. | M. J. A. d. Dood, A. Polmand, and J. G. Flerning, “Modified spontaneous emission from erbium-doped photonic layer-by-layer crystals,” Phys. Rev. B |

32. | K. Banaszek and P. L. Knight, “Quantum interference in three-photon down-conversion,” Phys. Rev. A |

**OCIS Codes**

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 28, 2004

Revised Manuscript: February 13, 2004

Published: March 8, 2004

**Citation**

Claudio Conti, Andrea Di Falco, and Gaetano Assanto, "Optical parametric oscillations in isotropic photonic crystals," Opt. Express **12**, 823-828 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-823

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### References

- S. Barland et al., �??Cavity solitons as pixels in semiconductor microcavities,�?? Nature 419, 699-702 (2002). [CrossRef] [PubMed]
- S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, �??Ultralow-threshold Raman laser using a spherical dielectric microcavity,�?? Nature 415, 621-623 (2002). [CrossRef] [PubMed]
- A. Yariv and W. H. Louisell, �??Theory of the Optical Parametric Oscillator,�?? IEEE J. Quantum Electron. QE-2, 418-424 (1966). [CrossRef]
- B. Crosignani, P. D. Porto, and A. Yariv, �??Coupled-mode theory and slowly-varying approximation in guided-wave optics,�?? Opt. Commun. 78, 237-239 (1990). [CrossRef]
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs N.J.,1984).
- E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
- S. John, �??Strong Localization of Photons in Certain Disordered Dielectric Superlattices,�?? Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Photonic Crystals (Princeton University Press, Princeton, 1995).
- K. Sakoda, Optical Properties of Photonic Crystals (Springer, Berlin, 2001).
- P. Tran, �??Photonic-band-structure calculation of material possessing Kerr nonlinearity,�?? Phys. Rev. B 52, 10673-10676 (1995). [CrossRef]
- V. Lousse and J. P. Vigneron, �??Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,�?? Phys. Rev. E 63, 027602 (2001). [CrossRef]
- S. F. Mingaleev and Y. S. Kivshar, �??Nonlinear transmission and light localization in photonic-crystal waveguides,�?? J. Opt. Soc. Am. B 19, 2241-2249 (2002). [CrossRef]
- M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, �??Optimal bistable switching in nonlinear photonic crystals,�?? Phys. Rev. E 66, 55601 (2002). [CrossRef]
- J. Vuckovic, O. Painter, Y. Xu, and A. Yariv, �??Finite-Difference Time-Domain Calculation of the Spontaneous Emission Coupling Factor in Optical Microcavities,�?? IEEE J. Quantum Electron. 35, 1168-1175 (1999). [CrossRef]
- K. Srinivasan and O. Painter, �??Momentum space design of high-Q photonic crystal optical cavities,�?? Optics Express 10, 670-684 (2002). <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a> [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego (CA), 1989).
- J.-P. Fève, B. Boulanger, J. Douady, �??Specific properties of cubic optical parametric interactions compared to quadratic interactions,�?? Phys. Rev. A 66, 063817 (2002). [CrossRef]
- M. Bahl, N.-C. Panoiu, and R. Osgood Jr., �??Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,�?? Phys. Rev. E 67, 56604 (2003). [CrossRef]
- Z. Y. Li and Y. Xia, �??Full vectorial model for quantum optics in three-dimensional photonic crystals,�?? Phys. Rev. A 63, 043817 (2001). [CrossRef]
- J. L. Young and R. O. Nelson, �??A Summary and Sistematic Analysis of FDTD Algorithms for Linearly Dispersive Media,�?? IEEE Antennas Propagat. Mag. 43, 61-77 (2001). [CrossRef]
- K. S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propagat. AP-14, 302-307 (1966).
- A. Taflove and S. C. Hagness, Computational Electrodynamics: the finite-difference time-domain method, 2 ed. (Artech House, London, 2000).
- J. Koga, �??Simulation model for the effects of nonlinear polarization on the propagation of intense pulse laser,�?? Opt. Lett. 24, 408-410 (1999). [CrossRef]
- R. W. Boyd, Nonlinear Optics, 2 ed. (Academic Press, New York, 2002).
- R. M. Joseph and A. Taflove, �??FDTD Maxwell�??s Equations Models for Nonlinear Electrodynamics and Optics,�?? IEEE Trans. Antennas Propagat. 45, 364-374 (1997). [CrossRef]
- S. John and K. Busch, �??Photonic Bandgap Formation and Tunability in Certain Self-Organizing Systems,�?? J. Lightwave Technol. 17, 1931-1943 (1999). [CrossRef]
- A. Blanco et al., �??Large-scale synthesis of a silicon photonic crystal with a complete three-dimensional bandgap near 1.5 micrometres,�?? Nature 405, 437-440 (2000). [CrossRef] [PubMed]
- J. S. Aitchison, D. C. Hutchings, J. U. Kang, G. I. Stegeman, and A. Villeneuve, �??The nonlinear optical properties of AlGaAs at the half band gap,�?? IEEE J. Quantum Electron. 33, 341-348 (1997). [CrossRef]
- R. Wang, X. H. Wang, B. Y. Gu, and G. Z. Yang, �??Local density of states in three-dimensional photonic crystals: Calculation and enhancement effects,�?? Phys. Rev. B 67, 155114 (2003). [CrossRef]
- R. W. Ziolkowski and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001). [CrossRef]
- M. J. A. d. Dood, A. Polmand, and J. G. Flerning, �??Modified spontaneous emission from erbium-doped photonic layer-by-layer crystals,�?? Phys. Rev. B 67, 115106 (2003). [CrossRef]
- K. Banaszek, and P. L. Knight, �??Quantum interference in three-photon down-conversion,�?? Phys. Rev. A 55, 2368 [CrossRef]

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